This paper is devoted to a detailed study of the randomized approximation of finite sums, i.e., sums ∑mj=1 xj, x ∈ ℝm, where m is supposed to be large, shall be approximated with information on n coordinates, only. The error is measured on balls in lmp, 1 ≤ p ≤ ∞. Main emphasis is laid on the exact solution of the problems stated below. In most cases we obtain both, an optimal method for the Monte Carlo setting and the description of least favorable distributions for the average case setting, exhibiting results obtained in a previous paper by the author, [Mat92]. Moreover, the solution of the finite-dimensional problem is applied to the Monte Carlo integration of continuous functions. Finally, this knowledge is used to study some of the properties, the optimal methods possess.